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Physical Review A

Generalized Rabi model in quantum-information processing including the A2 term

Liang-Hui Du, Xiang-Fa Zhou, Zheng-Wei Zhou, Xingxiang Zhou, Guang‐Can Guo

Published Jul 12, 2012• 14 citations
Publisher

Abstract

We consider the effect of the quadratic term ${\stackrel{P\vec}{A}}^{2}$ on quantum state transfer, nonclassical state preparation, and collapse-revival dynamics in the strong and ultrastrong coupling regime. Using the Bogoliubov transformation, we define a field mode which allows us to reduce the Hamiltonian to the original Rabi form without the ${\stackrel{P\vec}{A}}^{2}$ term. We calculate the error introduced by the ${\stackrel{P\vec}{A}}^{2}$ term in quantum state transfer and find it cannot be neglected in the strong coupling regime. We also study how it impacts the characteristics of the prepared nonclassical states and collapse-revival dynamics. The double-peak characteristic of the revival dynamics can be experimentally observed as manifestation of the effect of the ${\stackrel{P\vec}{A}}^{2}$ term.

DOI
10.1103/physreva.86.014303

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We consider the effect of the quadratic term {\stackrel{P\vec}{A}}^{2} on quantum state transfer, nonclassical state preparation, and collapse-revival dynamics in the strong and ultrastrong coupling regime. Using the Bogoliubov transformation, we define a field mode which allows us to reduce the Hamiltonian to the original Rabi form without the {\stackrel{P\vec}{A}}^{2} term. We calculate the error introduced by the {\stackrel{P\vec}{A}}^{2} term in quantum state transfer and find it cannot be neglected in the strong coupling regime. We also study how it impacts the characteristics of the prepared nonclassical states and collapse-revival dynamics. The double-peak characteristic of the revival dynamics can be experimentally observed as manifestation of the effect of the {\stackrel{P\vec}{A}}^{2} term.
The quantum Rabi model is in the scientific spotlight due to the recent theoretical and experimental progress. Nevertheless, a full-fledged classification of its coupling regimes remains as a relevant open question. We propose a spectral classification dividing the coupling regimes into three regions based on the validity of perturbative criteria on the quantum Rabi model, which allows us the use of exactly solvable effective Hamiltonians. These coupling regimes are (i) the perturbative ultrastrong coupling regime which comprises the Jaynes-Cummings model, (ii) a region where nonperturbative ultrastrong and nonperturbative deep strong coupling regimes coexist, and (iii) the perturbative deep strong coupling regime. We show that this spectral classification depends not only on the ratio between the coupling strength and the natural frequencies of the unperturbed parts, but also on the energy to which the system can access. These regimes additionally discriminate the completely different behaviors of several static physical properties, namely the total number of excitations, the photon statistics of the field, and the cavity-qubit entanglement. Finally, we explain the dynamical properties which are traditionally associated with the deep strong coupling regime, such as the collapses and revivals of the state population, in the frame of the proposed spectral classification.
The effects of counter-rotating terms (CRTs) on Rabi splitting and the dynamic evolution of atomic population in the Jaynes—Cummings model are studied with a coherent-state approach. When the coupling strength increases, the Rabi splitting becomes of multi-Rabi frequencies for the initial state of an excited atom in a vacuum field, and the collapses and revivals gradually disappear, and then reappear with quite good periodicity. Without the rotating-wave approximation (RWA), the initial excited state contains many eigenstates rather than two eigenstates under the RWA, which results in the multi-peak emission spectrum. An analytical approximate solution for the strong coupling regime is obtained, which gives a new oscillation frequency and explains the recovery of collapses and revivals due to the equal energy spacing.
The quantum Rabi model describes the fundamental mechanism of light-matter interaction. It consists of a two-level atom or qubit coupled to a quantized harmonic mode via a transversal interaction. In the weak coupling regime, it reduces to the well-known Jaynes-Cummings model by applying a rotating wave approximation. The rotating wave approximation breaks down in the ultra-strong coupling regime, where the effective coupling strength g is comparable to the energy ω of the bosonic mode, and remarkable features in the system dynamics are revealed. Here we demonstrate an analog quantum simulation of an effective quantum Rabi model in the ultra-strong coupling regime, achieving a relative coupling ratio of g/ω ~ 0.6. The quantum hardware of the simulator is a superconducting circuit embedded in a cQED setup. We observe fast and periodic quantum state collapses and revivals of the initial qubit state, being the most distinct signature of the synthesized model.An analog quantum simulation scheme has been explored with a quantum hardware based on a superconducting circuit. Here the authors investigate the time evolution of the quantum Rabi model at ultra-strong coupling conditions, which is synthesized by slowing down the system dynamics in an effective frame.
Abstract It is well known that the A-square term must be considered in both cavity and circuit quantum electrodynamics systems, because it arises in the derivation from the minimal coupling Hamiltonian at any finite coupling strength. In this paper, we study the quantum Rabi model with A-square terms using the Bogoliubov operator approach. After a unitary transformation, the A-square terms can be eliminated, resulting in a modified quantum Rabi model with renormalized parameters. A transcendental function responsible for the exact solution is then derived. The presence of the A-square terms is found to significantly alter the energy spectrum. The dynamics are also studied using the obtained exact wave function, which is sensitive to the strength of the A-square terms at strong coupling. We believe that these results could be observed in future light–matter interaction systems in the ultra-strong and deep strong coupling regimes.
We revisit the analytic solution of the two-qubit quantum Rabi model with inhomogeneous coupling and transition frequencies using a displaced oscillator basis. This approach enables us to apply the same truncation rules and techniques adopted in the Rabi model to the two qubits system. The derived analytical spectra match perfectly with the numerical solutions in the parameter regime where the qubits’ transition frequencies are far off-resonance with the field frequency and the interaction strengths reach the ultrastrong coupling regime. We further explore the dynamical behavior of the two qubits as well as the evolution of entanglement. The analytical methods provide unexpectedly accurate results in describing the dynamics of the two qubits in the present experimentally accessible coupling regime. The time evolutions of the probability for the qubits show that the collapse-revival phenomena emerge, survive and finally disappear when one coupling strength increases from weak to strong coupling regimes and the other coupling strength is well into the ultrastrong coupling regime. The inhomogeneous coupling system exhibits new dynamics, which are different from the homogeneous coupling case.
In the ultrastrong-coupling regime, the quantum Rabi model can exhibit quantum phase transition (QPT) when the ratio of the qubit transition frequency to the frequency of the cavity field approaches infinity. However, it is challenging to control the QPT in few-body systems because of the limited coupling strength and the A2{A}^{2}A2 term. Here, we propose a practical scheme to manipulate the QPT of quantum Rabi model in the strong-coupling regime. By applying a periodic frequency modulation to the two-level system in a quantum Rabi model in the strong-coupling regime, an anisotropic quantum Rabi model with ultrastrong and tunable coupling strengths for rotating and counterrotating terms is obtained. The ground state and excitation energy of this model, in terms of the modulation parameters, are studied. We find that the QPT of the quantum Rabi model can be observed in the strong-coupling regime and externally controlled by the modulation.